Find the general equation of a line perpendicular to $x + y + 4 = 0$ and passing through the point $(1, 2)$.

In what ratio does the line $y - x + 2 = 0$ divide the line segment joining the points $(3, -1)$ and $(8, 9)$?

Let the coordinates of the two points $A$ and $B$ be $(1, 2)$ and $(7, 5)$ respectively. The line $AB$ is rotated through $45^{\circ}$ in the anti-clockwise direction about the point of trisection of $AB$ which is nearer to $B$. The equation of the line in the new position is:

Find the values of $\theta$ and $p$,if the equation $x \cos \theta + y \sin \theta = p$ is the normal form of the line $\sqrt{3} x + y + 2 = 0$.

If a straight line is at a distance of $10$ units from the origin and the perpendicular drawn from the origin to it makes an angle of $\frac{\pi}{4}$ with the negative $X$-axis in the negative direction,then the equation of that line is:

Find the equation of the line whose perpendicular distance from the origin is $4$ units and the angle which the normal makes with the positive direction of the $x$-axis is $15^{\circ}$.